Einstein link spectra and the Morse index of Ricci-flat cones

Nifa, Anass

doi:10.5281/zenodo.22914003

Preprint · v1

Einstein link spectra and the Morse index of Ricci-flat cones

Anass Nifa3 September 2026

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Abstract

We study the low transverse-traceless spectrum of positive Einstein manifolds and its effect on the Morse index of Ricci-flat cones. Free quaternionic quotients of four equal round spheres have exactly determined first scalar and tensor eigenvalues. For four round two-spheres, the quotient cone is nine-dimensional, is stable and has restricted holonomy $SO(9)$ , whereas its eightfold cover has infinite negative index. For a general Einstein link, its invariant eigenspaces below the Hardy threshold determine the exact index on cone annuli and the leading logarithmic accumulation of negative eigenvalues on complete Ricci-flat manifolds with weighted $C^2$ convergence to the cone. No convergence rate is required for the leading term. At the threshold, an error of order $o((\log r)^{-2})$ makes cone stability equivalent to finite index. A pointwise comparison for canonically conformally Kähler Einstein four-manifolds gives, for the Chen–LeBrun–Weber metric at Einstein constant $3$ , the inequality $207/100<\sup D_k(s^2)<21/10$ , where $g=s^{-2}k$ , $s=\operatorname{Scal}_k>0$ and $D_k=\operatorname{div}_k\nabla$ . This proves the Hall–Haslhofer–Siepmann derivative inequality and improves the low tensor eigenvalue bound. Applications include an explicit leading coefficient for the complete Ricci-flat Böhm metrics, an index formula on long Einstein necks, and Euclidean rigidity for finite-index five-dimensional fillings under a uniform half-Weyl eigenvalue inequality.

MSC 2020

53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
58J50: Spectral problems; spectral geometry; scattering theory on manifolds
35P20: Asymptotic distributions of eigenvalues in context of PDEs
53C55: Global differential geometry of Hermitian and Kählerian manifolds
53E20: Ricci flows

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22914003
All versions: 10.5281/zenodo.22914002
Subjects: Einstein metric; Ricci-flat cone; Lichnerowicz operator; Morse index; finite quotient; conformally Kähler metric; inverse-square potential
Language: English

Cite this paper

Anass Nifa. Einstein link spectra and the Morse index of Ricci-flat cones. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22914003

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References

As listed in the manuscript bibliography.

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M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333–340.
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D. N. Page, A compact rotating gravitational instanton, Phys. Lett. B 79 (1978), no. 3, 235–238.
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G. Verger, Instability of Böhm's Einstein metrics, arXiv:2608.25865v1, 26 August 2026, preprint.
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[27]
L. Yudowitz, Semi-Continuity of the Morse Index for Ricci Shrinkers, J. Geom. Anal. 35 (2025), article 159; doi:10.1007/s12220-025-01999-1. Numbered citations refer to arXiv:2408.10751v2.
DOI · arXiv v2

[reference-1] [1]
W. Ambrose and I. M. Singer, A theorem on holonomy, Trans. Amer. Math. Soc. 75 (1953), 428–443.

[reference-2] [2]
S. B. Angenent and D. Knopf, Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions, arXiv:2503.12210v1, 15 March 2025, preprint.
arXiv v1

[reference-3] [3]
A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10, Springer-Verlag, Berlin, 1987.

[reference-4] [4]
O. Biquard and T. Ozuch, Instability of conformally Kähler, Einstein metrics, J. Differential Geom. 133 (2026), no. 3, 335–346; doi:10.4310/jdg/1779980240. Numbered citations refer to arXiv:2310.10109v3.
DOI · arXiv v3

[reference-5] [5]
C. Böhm, Non-compact cohomogeneity one Einstein manifolds, Bull. Soc. Math. France 127 (1999), no. 1, 135–177; doi:10.24033/bsmf.2345.
DOI

[reference-6] [6]
G. Catino and D. Dameno, Conformal Kähler rigidity of Einstein four-manifolds, arXiv:2607.21538v1, 23 July 2026, preprint.
arXiv v1

[reference-7] [7]
X. Chen, C. LeBrun and B. Weber, On conformally Kähler, Einstein manifolds, J. Amer. Math. Soc. 21 (2008), no. 4, 1137–1168; arXiv:0705.0710v2.
arXiv v2

[reference-8] [8]
A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), no. 3, 405–433.

[reference-9] [9]
P. B. Gilkey, Invariance theory, the heat equation, and the Atiyah–Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Wilmington, DE, 1984; Sections 1.3 and 1.6.

[reference-10] [10]
S. Hall, R. Haslhofer and M. Siepmann, The stability inequality for Ricci-flat cones, J. Geom. Anal. 24 (2014), no. 1, 472–494; arXiv:1111.4981v1, 21 November 2011. Page and equation locators refer to this preprint.
arXiv v1

[reference-11] [11]
S. J. Hall and T. Murphy, Numerical approximations to extremal toric Kähler metrics with arbitrary Kähler class, Proc. Edinb. Math. Soc. (2) 60 (2017), no. 4, 893–910; arXiv:1407.1272v2. Numbered citations, including Conjecture 5.3, refer to version 2.
arXiv v2

[reference-12] [12]
A. Hassell and S. Marshall, Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree $-2$, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4145–4167; arXiv:math/0510617.
arXiv

[reference-13] [13]
W. Kirsch and B. Simon, Corrections to the classical behavior of the number of bound states of Schrödinger operators, Ann. Physics 183 (1988), no. 1, 122–130.

[reference-14] [14]
K. Kröncke, On infinitesimal Einstein deformations, Differential Geom. Appl. 38 (2015), 41–57; arXiv:1508.00721v1. The product decomposition used here is Section 4 of this version.
arXiv v1

[reference-15] [15]
K. Kröncke, Stable and unstable Einstein warped products, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6537–6563; arXiv:1507.01782.
arXiv

[reference-16] [16]
K. Kröncke and Á. Szabó, Optimal coordinates for Ricci-flat conifolds, Calc. Var. Partial Differential Equations 63 (2024), no. 7, article 188, 41 pp.; doi:10.1007/s00526-024-02780-y; arXiv:2203.01711v1. Numbered citations refer to the preprint, in particular Theorem 3.15.
DOI · arXiv v1

[reference-17] [17]
C. LeBrun, Einstein metrics, conformal curvature, and anti-holomorphic involutions, Ann. Math. Québec 45 (2021), 391–405; arXiv:2007.01180.
arXiv

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C. LeBrun, On Einstein, Hermitian four-manifolds, J. Differential Geom. 90 (2012), no. 2, 277–302; arXiv:1010.0238.
arXiv

[reference-19] [19]
A. Nifa, Negative Lichnerowicz modes on Einstein warped products and spheres, preprint, Zenodo, version 1, 2026, 21 pp.; doi:10.5281/zenodo.22681024.
DOI

[reference-20] [20]
A. Nifa, Scalar and tensor eigenvalue bounds for conformally Kähler Einstein four-manifolds, preprint, 2026, 31 pp.; doi:10.5281/zenodo.22851442.
DOI

[reference-21] [21]
M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333–340.

[reference-22] [22]
D. N. Page, A compact rotating gravitational instanton, Phys. Lett. B 79 (1978), no. 3, 235–238.

[reference-23] [23]
M. Reed and B. Simon, Methods of modern mathematical physics IV: Analysis of operators, Academic Press, New York, 1978; Chapter XIII.

[reference-24] [24]
P. Schwahn, Stability of Einstein metrics on symmetric spaces of compact type, Ann. Global Anal. Geom. 61 (2022), 333–357; doi:10.1007/s10455-021-09810-4.
DOI

[reference-25] [25]
P. Schwahn and U. Semmelmann, Einstein metrics, their moduli spaces and stability, arXiv:2507.18463v3, 2026.
arXiv v3

[reference-26] [26]
G. Verger, Instability of Böhm's Einstein metrics, arXiv:2608.25865v1, 26 August 2026, preprint.
arXiv v1

[reference-27] [27]
L. Yudowitz, Semi-Continuity of the Morse Index for Ricci Shrinkers, J. Geom. Anal. 35 (2025), article 159; doi:10.1007/s12220-025-01999-1. Numbered citations refer to arXiv:2408.10751v2.
DOI · arXiv v2